Optimal. Leaf size=35 \[ \frac{(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0412381, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 11.1162, size = 29, normalized size = 0.83 \[ - \frac{\left (a e + c d x\right )^{2}}{2 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.0205752, size = 29, normalized size = 0.83 \[ -\frac{a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]
[Out]
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Maple [A] time = 0.009, size = 40, normalized size = 1.1 \[ -{\frac{a{e}^{2}-c{d}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{cd}{{e}^{2} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.723461, size = 58, normalized size = 1.66 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201392, size = 58, normalized size = 1.66 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.77239, size = 44, normalized size = 1.26 \[ - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.206132, size = 62, normalized size = 1.77 \[ -\frac{{\left (2 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3} + a x e^{3} + a d e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="giac")
[Out]